A regional telephone company is analyzing the number of telephone calls that are connected to wrong numbers at its telephone exchange. It collects the number of wrong telephone connections on each of 200 days, and treats the observations as the outcome of a random sample of size 200 from a Poisson population distribution:

\(f(z ; \lambda)=\frac{e^{-\lambda} \lambda^{z}}{z !} I_{\{0,1,2, \ldots\}}(z)\)

(a) Define the MLE of \(\lambda\), the expected number of wrong connections per day.

(b) Is the MLE the MVUE for \(\lambda\) ? Is it consistent? Asymptotically normal? Asymptotically efficient?

(c) If \(\sum_{i=1}^{200} \mathbf{x}_{i}=4,973\), what is the ML estimate of the expected number of wrong connections? If each wrong connection costs the company \(\$ 070\), define a MLE for the expected daily cost of wrong connections, and generate a ML estimate of this cost.

(d) Define a MLE for the standard deviation of the daily number of wrong connections. Is the MLE consistent? Is it asymptotically normal? Generate a ML estimate of the standard deviation.